beauty of complex geometry

Algebra Level pending

Let |z1 -4i -a(1 -i) |=|a(1-i)+(-1+3i)| , (a €R) there exist exactly two distinct complex no z, which. also satisfy the locus arg(Z2)=45°

If z3 and z4 be two complex no.s that satisfy the locus of both. z1 and z2. find |z3 -z4|

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